3.1.69 \(\int \frac {a+b x}{(2-x^2) \sqrt [4]{-1+x^2}} \, dx\) [69]

3.1.69.1 Optimal result

Integrand size = 24, antiderivative size = 80 \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-b \arctan \left (\sqrt [4]{-1+x^2}\right )+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}+b \text {arctanh}\left (\sqrt [4]{-1+x^2}\right ) \]

output

-b*arctan((x^2-1)^(1/4))+b*arctanh((x^2-1)^(1/4))+1/4*a*arctan(1/2*x/(x^2- 
1)^(1/4)*2^(1/2))*2^(1/2)+1/4*a*arctanh(1/2*x/(x^2-1)^(1/4)*2^(1/2))*2^(1/ 
2)
 

3.1.69.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.96 \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\frac {x \left (b x \sqrt [4]{1-x^2} \left (-2+x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,x^2,\frac {x^2}{2}\right )-\frac {24 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )}{6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},x^2,\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},x^2,\frac {x^2}{2}\right )\right )}\right )}{4 \left (-2+x^2\right ) \sqrt [4]{-1+x^2}} \]

input

Integrate[(a + b*x)/((2 - x^2)*(-1 + x^2)^(1/4)),x]
 

output

(x*(b*x*(1 - x^2)^(1/4)*(-2 + x^2)*AppellF1[1, 1/4, 1, 2, x^2, x^2/2] - (2 
4*a*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2])/(6*AppellF1[1/2, 1/4, 1, 3/2, 
x^2, x^2/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, x^2, x^2/2] + AppellF1[3/2 
, 5/4, 1, 5/2, x^2, x^2/2]))))/(4*(-2 + x^2)*(-1 + x^2)^(1/4))
 

3.1.69.3 Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1343, 309, 353, 73, 827, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}} \, dx\)

\(\Big \downarrow \) 1343

\(\displaystyle a \int \frac {1}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx+b \int \frac {x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx\)

\(\Big \downarrow \) 309

\(\displaystyle b \int \frac {x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {1}{2} b \int \frac {1}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx^2+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle 2 b \int \frac {x^4}{1-x^8}d\sqrt [4]{x^2-1}+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 827

\(\displaystyle 2 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{x^2-1}-\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{x^2-1}\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle 2 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{x^2-1}-\frac {1}{2} \arctan \left (\sqrt [4]{x^2-1}\right )\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )+2 b \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{x^2-1}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{x^2-1}\right )\right )\)

input

Int[(a + b*x)/((2 - x^2)*(-1 + x^2)^(1/4)),x]
 

output

a*(ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/(2*Sqrt[2]) + ArcTanh[x/(Sqrt[2]*( 
-1 + x^2)^(1/4))]/(2*Sqrt[2])) + 2*b*(-1/2*ArcTan[(-1 + x^2)^(1/4)] + ArcT 
anh[(-1 + x^2)^(1/4)]/2)
 

3.1.69.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 309

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + 
 b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2]*(a 
+ b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && 
NegQ[b^2/a]
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1343

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q 
_), x_Symbol] :> Simp[g   Int[(a + c*x^2)^p*(d + f*x^2)^q, x], x] + Simp[h 
  Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h, p, 
q}, x]
 

3.1.69.4 Maple [F]

\[\int \frac {b x +a}{\left (-x^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}}d x\]

input

int((b*x+a)/(-x^2+2)/(x^2-1)^(1/4),x)
 

output

int((b*x+a)/(-x^2+2)/(x^2-1)^(1/4),x)
 

3.1.69.5 Fricas [F(-1)]

Timed out. \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\text {Timed out} \]

input

integrate((b*x+a)/(-x^2+2)/(x^2-1)^(1/4),x, algorithm="fricas")
 

output

Timed out
 

3.1.69.6 Sympy [F]

\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=- \int \frac {a}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx - \int \frac {b x}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx \]

input

integrate((b*x+a)/(-x**2+2)/(x**2-1)**(1/4),x)
 

output

-Integral(a/(x**2*(x**2 - 1)**(1/4) - 2*(x**2 - 1)**(1/4)), x) - Integral( 
b*x/(x**2*(x**2 - 1)**(1/4) - 2*(x**2 - 1)**(1/4)), x)
 

3.1.69.7 Maxima [F]

\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}} \,d x } \]

input

integrate((b*x+a)/(-x^2+2)/(x^2-1)^(1/4),x, algorithm="maxima")
 

output

-integrate((b*x + a)/((x^2 - 1)^(1/4)*(x^2 - 2)), x)
 

3.1.69.8 Giac [F]

\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}} \,d x } \]

input

integrate((b*x+a)/(-x^2+2)/(x^2-1)^(1/4),x, algorithm="giac")
 

output

integrate(-(b*x + a)/((x^2 - 1)^(1/4)*(x^2 - 2)), x)
 

3.1.69.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int -\frac {a+b\,x}{{\left (x^2-1\right )}^{1/4}\,\left (x^2-2\right )} \,d x \]

input

int(-(a + b*x)/((x^2 - 1)^(1/4)*(x^2 - 2)),x)
 

output

int(-(a + b*x)/((x^2 - 1)^(1/4)*(x^2 - 2)), x)
 

3.1.69.10 Reduce [F]

\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=-\left (\int \frac {x}{\left (x^{2}-1\right )^{\frac {1}{4}} x^{2}-2 \left (x^{2}-1\right )^{\frac {1}{4}}}d x \right ) b -\left (\int \frac {1}{\left (x^{2}-1\right )^{\frac {1}{4}} x^{2}-2 \left (x^{2}-1\right )^{\frac {1}{4}}}d x \right ) a \]

input

int(( - (a + b*x))/((x**2 - 1)**(1/4)*(x**2 - 2)),x)
 

output

 - (int(x/((x**2 - 1)**(1/4)*x**2 - 2*(x**2 - 1)**(1/4)),x)*b + int(1/((x* 
*2 - 1)**(1/4)*x**2 - 2*(x**2 - 1)**(1/4)),x)*a)